Find the inverse of the function on the given domain. f-1(x) = f(x)=(x-4)2, [4,00)

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Finding the Inverse of a Function

**Problem Statement:**
Find the inverse of the function on the given domain.
\[ f(x) = (x - 4)^2, \quad [4, \infty) \]

The solution needs to determine \( f^{-1}(x) \).

**Detailed Explanation:**

1. **Function Provided:**
   \[ f(x) = (x - 4)^2 \]

   Notice the domain restriction: \([4, \infty)\). This means the function \( f(x) \) is defined for \( x \geq 4 \).

2. **Steps to Find the Inverse Function:**
   - Swap \( x \) and \( y \): 
     \[ y = (x - 4)^2 \]
     becomes
     \[ x = (y - 4)^2 \]

   - Solve for \( y \):
     \[ \sqrt{x} = y - 4 \]
     (We take the positive square root because \( x \geq 4 \))
     \[ y = \sqrt{x} + 4 \]

3. **Conclusion:**
   The inverse of the given function \( f(x) = (x - 4)^2 \) on the domain \([4, \infty)\) is:
   \[ f^{-1}(x) = \sqrt{x} + 4 \]

   Therefore:
   \[ f^{-1}(x) = \underline{\sqrt{x} + 4} \]
   
4. **Additional Notes:**
   Ensure that all operations are within the specified domain to maintain validity and avoid extraneous solutions.
Transcribed Image Text:### Finding the Inverse of a Function **Problem Statement:** Find the inverse of the function on the given domain. \[ f(x) = (x - 4)^2, \quad [4, \infty) \] The solution needs to determine \( f^{-1}(x) \). **Detailed Explanation:** 1. **Function Provided:** \[ f(x) = (x - 4)^2 \] Notice the domain restriction: \([4, \infty)\). This means the function \( f(x) \) is defined for \( x \geq 4 \). 2. **Steps to Find the Inverse Function:** - Swap \( x \) and \( y \): \[ y = (x - 4)^2 \] becomes \[ x = (y - 4)^2 \] - Solve for \( y \): \[ \sqrt{x} = y - 4 \] (We take the positive square root because \( x \geq 4 \)) \[ y = \sqrt{x} + 4 \] 3. **Conclusion:** The inverse of the given function \( f(x) = (x - 4)^2 \) on the domain \([4, \infty)\) is: \[ f^{-1}(x) = \sqrt{x} + 4 \] Therefore: \[ f^{-1}(x) = \underline{\sqrt{x} + 4} \] 4. **Additional Notes:** Ensure that all operations are within the specified domain to maintain validity and avoid extraneous solutions.
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